Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | #45801 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4372 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,8)(3,7,4,6)(9,10,14,17)(11,18,13,15)(12,20,21,16)(19,23,22,24)(25,26)(27,30)(28,31)(29,32)(33,37)(34,38)(35,39)(36,40);(1,3,5,4)(2,6,8,7)(9,11,14,13)(10,15,17,18)(12,19,21,22)(16,23,20,24)(25,27,29,34)(26,30,32,38)(28,35,36,33)(31,39,40,37);(1,3,5,4)(2,7,8,6)(9,12,14,21)(10,16,17,20)(11,19,13,22)(15,23,18,24)(25,28)(26,31)(27,33)(29,36)(30,37)(32,40)(34,35)(38,39);(1,3,5,4)(2,7,8,6)(9,13,14,11)(10,15,17,18)(12,19,21,22)(16,24,20,23)(26,32)(28,36)(30,38)(33,35);(10,17)(15,18)(16,20)(23,24)(28,36)(31,40)(33,35)(37,39);(1,4,5,3)(2,6,8,7)Polycyclic presentation:
2^9|a^2=g,b^2=gi,c^2=g,d^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dhi,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7609717);